Theorem ptbasid

Description: The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015)

	Ref	Expression
Hypothesis	ptbas.1	⊢ 𝐵 = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) }
Assertion	ptbasid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ptbas.1	⊢ 𝐵 = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) }
2		simpl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐴 ∈ 𝑉 )
3		0fin	⊢ ∅ ∈ Fin
4	3	a1i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∅ ∈ Fin )
5		ffvelrn	⊢ ( ( 𝐹 : 𝐴 ⟶ Top ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑘 ) ∈ Top )
6	5	adantll	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑘 ) ∈ Top )
7		eqid	⊢ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 )
8	7	topopn	⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ Top → ∪ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝐹 ‘ 𝑘 ) )
9	6 8	syl	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ) → ∪ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝐹 ‘ 𝑘 ) )
10		eqidd	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ ( 𝐴 ∖ ∅ ) ) → ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) )
11	1 2 4 9 10	elptr2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∈ 𝐵 )